Piecewise Polynomial Model of the Aerodynamic Coefficients of the Cumulus One Unmanned Aircraft

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Piecewise Polynomial Model of the Aerodynamic

Coeﬀicients of the Cumulus One Unmanned Aircraft

Torbjørn Cunis, Anders La Cour-Harbo

To cite this version:

Torbjørn Cunis, Anders La Cour-Harbo. Piecewise Polynomial Model of the Aerodynamic Coeﬀicients

of the Cumulus One Unmanned Aircraft. [Technical Report] SkyWatch A/S, Støvring, DK. 2019.

�hal-02280789�

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Piecewise Polynomial Model of the Aerodynamic Coefficients of

the Cumulus One Unmanned Aircraft

Torbjørn Cunis

1

and Anders La Cour-Harbo

2

Abstract

This document illustrates a piecewise polynomial model for Cumulus One, based on continuous fluid dynamics simulation, derived using the pwpfit toolbox. The model has been developed in a joint project of Sky-Watch and the University of Aalborg.

Preliminaries

If not stated otherwise, all variables are in SI units. We will refer to the following axis systems of ISO 1151-1: the body axis system (xf, y_f, z_f) aligned with the aircraft’s fuselage; the air-path axis system (x_a, y_a, z_a) defined by the velocity vector VA; and the normal earth-fixed axis system (xg, y_g, z_g). The orientation of the body axes with respect to the normal earth-fixed system is given by the attitude angles Φ, Θ, Ψ and to the air-path system by angle of attack α and side-slip β; the orientation of the air-path axes to the normal earth-fixed system is given by azimuth χA, inclination γA, and bank-angle µA. (Fig. ??.)

xg xa x_f zg z_a z_f mg −ZA F −XA V_A Θ γA α

(a) Longitudinal axes (β = µA= 0).

x_g x0_f xa y_g y 0 f _y a F0 −XA YA0 V_A Ψ β0 χ_A (b) Horizontal axes (γA= 0).

Fig. 1: Axis systems with angles and vectors. Projections into the plane are marked by0.

I. Aerodynamic Coefficients The piece-wise polynomial models of the aerodynamic coefficients are given

C(α, β, ξ, η, ζ) =

Cpre(α, β, . . . ) if α ≤ α0,

Cpost(α, β, . . . ) else, (1) where C ∈ {CX, CY, CZ, Cl, Cm, Cn} are polynomials in angle of attack, side-slip, and surface deflections; and the

boundary is found at

α0=17.949°. (2)

The polynomials in low and high angle of attack, Cpre

, Cpost, are sums

Cpre = C_αpre(α) + Cξ(β, ξ) + Cηpre(α, η) + Cζ(β, ζ) ; (3)

Cpost= C_αpost(α) + Cξ(β, ξ) + Cηpost(α, η) + Cζ(β, ζ) . (4)

In the following subsections, we present the polynomial terms obtained using the pwpfit toolbox. Coefficients of absolute value lower than 10−2 _{have been omitted for readability.}

1_{ONERA – The French Aerospace Lab, Department of Information Processing and Systems, Centre Midi-Pyrénées, Toulouse, 31055,} France, e-mail:torbjoern.cunis@onera.fr; and ENAC, Université de Toulouse, Drones Research Group, Toulouse, 31055, France.

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A. Domain of low angle of attack

Polynomials in angle of attack:

C_Xαpre= −2.566 × 10−2+5.722 × 10−1α +1.496α2−1.148 × 101α3; (5) C_Yαpre=5.402 × 10−2−2.345 × 10−1α −2.001α2+7.054α3; (6) C_Zαpre= −3.475 × 10−1−5.467α + 1.853α2₊_{2.663 × 10}1_α3_; ₍₇₎ C_lαpre=4.875 × 10−2−2.190 × 10−1α −2.004α2+7.146α3; (8) C_mαpre=6.214 × 10−2−1.755α − 3.427α2+1.256 × 101α3; (9) C_nαpre=4.748 × 10−2−2.097 × 10−1α −2.016α2+7.171α3; (10) Polynomials in angle of attack and elevator deflections:

C_Xηpre=4.327 × 10−2η −4.458 × 10−1αη +3.370 × 10−1α2η −4.567 × 10−1αη2+7.331 × 10−2η3; (11) C_Yηpre=1.832 × 10−2η +7.484 × 10−2αη −4.384 × 10−1α2η; (12) C_Zηpre= −2.567 × 10−1_{η +}_{3.085 × 10}−1_{αη −}_{5.105 × 10}−2_η2₋7.394 × 10−1_α2_{η +}6.936 × 10−1_αη2₊1.337 × 10−1_η3_; (13) C_lηpre=1.699 × 10−2η +8.936 × 10−2αη −4.564 × 10−1α2η +1.571 × 10−2αη2; (14) C_mηpre= −9.028 × 10−1η +7.437 × 10−1αη −4.924 × 10−2η2−8.415 × 10−1α2η +2.210αη2+5.251 × 10−1η3; (15) C_nηpre=1.532 × 10−2η +8.758 × 10−2αη −4.513 × 10−1α2η +1.487 × 10−2αη2. (16)

B. Domain of high angle of attack

Polynomials in angle of attack:

C_Xαpost=1.266 × 10−2−3.159 × 10−1α +3.832 × 10−1α2−1.226 × 10−1α3; (17) C_Yαpost= −2.297 × 10−2α2+1.337 × 10−2α3; (18) C_Zαpost= −4.179 × 10−1−2.345α + 9.586 × 10−1α2−3.665 × 10−2α3; (19) C_lαpost=2.006 × 10−2−8.200 × 10−2α +1.012 × 10−1α2−3.515 × 10−2α3; (20) C_mαpost= −2.552 × 10−1−5.131 × 10−1α −2.677 × 10−1α2+1.332 × 10−1α3; (21) C_nαpost=1.159 × 10−2−3.062 × 10−2α +2.727 × 10−2α2; (22) Polynomials in angle of attack and elevator deflections:

C_Xηpost= −1.342 × 10−2_{η −}_{1.663 × 10}−1_{αη −}_{1.796 × 10}−1_η2₊2.254 × 10−2_α2_{η +}9.274 × 10−2_αη2₊7.331 × 10−2_η3_; (23) C_Zηpost= −2.922 × 10−1_{η +}_{1.829 × 10}−1_{αη +}_{1.277 × 10}−1_η2₊2.284 × 10−2_α2_{η +}1.229 × 10−1_αη2₊1.337 × 10−1_η3_; (24) C_mηpost= −9.498 × 10−1η +6.099 × 10−1αη +5.093 × 10−1η2₊6.456 × 10−2_α2_{η +}4.264 × 10−1_αη2₊5.251 × 10−1_η3_; (25) C_Yηpost, C_lηpost, C_nηpost are approximately zero. (26)

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C. Full-envelope polynomials

Polynomials in side-slip and aileron deflections: C_Xξ= 6.557 × 10−2_β2₊_{4.214 × 10}−2_{βξ −}_1.493ξ2₊_{4.264 × 10}−2_ξ3₋_{1.647 × 10}−2_β4₋_{2.321 × 10}−2_β3_{ξ +}_{9.649 × 10}−2_β2_ξ2₊_{2.859 × 10}−2_βξ3₊_{3.084 × 10}1_ξ4_; (27) CYξ= −3.697 × 10−1_{β −}_{1.570 × 10}−1_{ξ −}_{3.231 × 10}−2_β2₋_6.137ξ2₊_{7.416 × 10}−2_β3₊_{1.611 × 10}−2_βξ2₊_2.214ξ3₊_{6.487 × 10}−1_β2_ξ2₊_{5.323 × 10}−2_βξ3₊_{1.456 × 10}2_ξ4_; (28) C_Zξ= 3.411 × 10−1_β2₊4.141 × 10−1_{βξ −}_3.717ξ2₊4.234 × 10−2_ξ3₋9.915 × 10−2_β4₋1.922 × 10−1_β3_{ξ +}1.945 × 10−1_β2_ξ2₊9.909 × 10−1_βξ3₊9.856 × 101_ξ4_; (29) C_lξ= −5.798 × 10−2_{β −}_{3.929 × 10}−1_{ξ −}_{2.906 × 10}−2_β2₋_5.551ξ2₊_{1.763 × 10}−2_β3₊_{1.722 × 10}−1_β2_{ξ +}_{4.557 × 10}−1_ξ3₊_{5.835 × 10}−1_β2_ξ2₊_{5.332 × 10}−2_βξ3₊_{1.319 × 10}2_ξ4_; (30) C_mξ= −3.978 × 10−2β2₊_{5.554 × 10}−1_{βξ −}_4.689ξ2₊_{4.229 × 10}−2_ξ3₊_{2.585 × 10}−2_β4₋_{2.309 × 10}−1_β3_{ξ +}_{4.077 × 10}−1_β2_ξ2₋_{5.068 × 10}−1_βξ3₊_{1.160 × 10}2_ξ4_; (31) Cnξ= 3.686 × 10−2_{β +}_{1.392 × 10}−2_{ξ −}_{2.828 × 10}−2_β2₋_5.410ξ2₋_{1.160 × 10}−1_ξ3₊_{5.678 × 10}−1_β2_ξ2₊_{5.334 × 10}−2_βξ3₊_{1.286 × 10}2_ξ4_. (32) Polynomials in side-slip and rudder deflections:

CXζ = −1.031 × 10−2βζ −7.986 × 10−2ζ2+1.086 × 10−2βζ3+9.739 × 10−2ζ4; (33) C_Yζ = −8.526 × 10−2_{ζ −}_{2.128 × 10}−2_β2₋3.569 × 10−1_ζ2₊1.049 × 10−2_β2_{ζ +}1.924 × 10−2_βζ2₊7.679 × 10−2_ζ3₊5.663 × 10−2_β2_ζ2₊5.247 × 10−1_ζ4_; (34) CZζ= −2.036 × 10−2β2−1.332 × 10−2βζ −1.743 × 10−1ζ2+1.834 × 10−2β2ζ2+3.683 × 10−2βζ3+2.631 × 10−1ζ4; (35) Clζ= −1.930 × 10−2β2−3.221 × 10−1ζ2+5.111 × 10−2β2ζ2+4.735 × 10−1ζ4; (36) C_mζ = −3.231 × 10−2_β2₋1.436 × 10−2_{βζ −}2.672 × 10−1_ζ2₊1.241 × 10−2_β3_{ζ +}1.472 × 10−2_β2_ζ2₊1.051 × 10−2_βζ3₊4.193 × 10−1_ζ4_; (37) C_nζ=2.158 × 10−2ζ −1.882 × 10−2β2−3.137 × 10−1ζ2−1.338 × 10−2ζ3+4.978 × 10−2β2ζ2+4.611 × 10−1ζ4. (38) II. Equations of Motion

We have the aerodynamic forces and moments in body axis system   XA YA ZA   f = 1 2%SV 2 A   CX CY CZ  ;   LA MA NA   f = 1 2%SV 2 A   b Cl cACm b Cn  +   XA YA ZA   f × xcg− xref_cg ; (39)

and the weight force by rotation into the body axis system

X_gG= −gsin Θ; (40) Y_gG= −gsin Φ cos Θ; (41) Z_gG= −gcos Φ cos Θ; (42) The resulting forces lead to changes in the velocity vector, in body axis system, by

˙ V_Af= 1 m   XA+ XG+ XF YA+ YG ZA+ ZG   f +   p q r  × VAf. (43)

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with the thrust XF

f = F (engines aligned with the xf-axis). For a symmetric plane (Ixy = Iyz = 0), the resulting

moments in body axis are given as

L_f= LA_f − q r (Iz− Iy) + p q Izx; (44)

Mf= M_fA− p r (Ix− Iz) − p2− r2 Izx; (45)

N_f= N_fA− p q (Iy− Ix) − q r Izx; (46)

with the inertias Ix, Iy, Iz, Izx. The changes of angular body rates are then given as

˙ p = 1 IxIz− Izx2 (IzLf+ IzxNf) ; (47) ˙ q = 1 Iy M_f; (48) ˙r = 1 IxIz− Izx2 (IzxLf+ IxNf) . (49)

Here, the normalized body rates have been used with   ˆ p ˆ q ˆ r  = 1 2V_A   b p cAq b r  ⇐⇒   p q r  = 2VA   b−1pˆ c−1_A qˆ b−1rˆ   (50) and   ˙ˆ p ˙ˆq ˙ˆr  = 1 V_A   1 2   b ˙p c_Aq˙ b ˙r  − ˙VA   ˆ p ˆ q ˆ r    . (51)

The change of attitude is finally obtained by rotation into normal earth-fixed axis system: ˙

Φ = p + qsin Φ tan Θ + r cos Φ tan Θ; (52) ˙

Θ = qcos Φ − r sin Φ; (53) ˙

Ψ = qsin Φ cos−1Θ + rcos Φ cos−1Θ. (54) III. Longitudinal Model

The longitudinal model is restricted to the xa-za-plane, assuming β = µA= χ_A= 0.

A. Aerodynamic Coefficients

The longitudinal aerodynamic coefficients, CL, C_D, C_m, are obtained as C_L C_D = +sin α − cos α −cos α − sin α C_X C_Z . (55) B. Equations of motion

The longitudinal equations of motion are given as ˙ VA= 1 m Tcos α −1 2ρSV 2 ACD(α, η, q) − mgsin γA , (56) ˙γ_A= 1 mV_A Tsin α + 1 2ρSV 2 ACL(α, η, q) − mgcos γ_A , (57) ˙ q = 1 Iy 1 2ρScAV 2 ACm(α, η, q) ; (58) ˙ Θ = q; (59) with Θ = α + γ. (60)

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Nomenclature α = Angle of attack (rad);

α0 = Low-angle of attack boundary (°);

β = Side-slip angle (rad);

γA = Flight-path angle relative to air (rad);

ζ = Rudder deflection (rad), negative if leading to positive yaw moment; η = Elevator deflection (rad), negative if leading to positive pitch moment; Θ = Pitch angle (rad);

µ_A = Air-path bank angle (rad);

ξ = Aileron deflection (rad), negative if leading to positive roll moment; % = Air density (% = 1.200 kg m−3);

χ_A = Air-path azimuth angle (rad); Φ = Bank angle (rad);

Ψ = Azimuth angle (rad);

Cl = Aerodynamic coefficient moment body xf-axis (·);

Cm = Aerodynamic coefficient moment body yf-axis (·);

Cn = Aerodynamic coefficient moment body zf-axis (·);

CD = Aerodynamic drag coefficient (·), positive along negative air-path xa-axis;

CL = Aerodynamic lift coefficient (·), positive along negative air-path za-axis;

C_X = Aerodynamic coefficient force body xf-axis (·);

C_Y = Aerodynamic coefficient force body yf-axis (·);

C_Z = Aerodynamic coefficient force body zf-axis (·);

D = Drag force (N), positive along negative air-path xa-axis;

F = Thrust force (N), positive along body xf-axis;

L = Lift force (N), positive along negative air-path za-axis;

q = Pitch rate (rad s−1);

b = Reference aerodynamic span (b = 2.088 m); c_A = Aerodynamic mean chord (c_A=2.800 × 10−1m); g = Standard gravitational acceleration (g ≈ 9.810 m s−2); m = Aircraft mass (m = 2.619 × 101kg);

p = Roll rate (rad s−1); r = Yaw rate (rad s−1);

Lf = Roll moment (N m), mathematically positive around xf-axis;

Mf = Pitch moment (N m), mathematically positive around yf-axis;

N_f = Yaw moment (N m), mathematically positive around z_f-axis; S = Wing area (S = 5.500 × 10−1m2);

V_A, V_A = Aircraft speed and velocity relative to air (V_A= kV_Ak₂, m s−1); X_fA = Aerodynamic force along body x_f-axis (N);

X_fF = Thrust force along body x_f-axis (N); Y_fA = Aerodynamic force along body y_f-axis (N); Z_fA = Aerodynamic force along body z_f-axis (N); (·)post = Domain of high angle of attack;

(·)pre = Domain of low angle of attack; xa, ya, za = Air-path axis system;

xf, yf, zf = Body axis system;